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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Suppose that the initial condition is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_1.html ./knowl/eq8_3.html">
\begin{equation*}
{\bf x}(t_0)=\left(
\begin{array}{c}
x_1^0\\
x_2^0\\
\vdots\\
x_n^0
\end{array}
\right), \quad \alpha \leq t \leq \beta.
\end{equation*}
</div>
<p class="continuation">In (<a href="" class="xref" data-knowl="./knowl/eq8_1.html" title="Equation 6.5.1">(6.5.1)</a>), let <span class="process-math">\(t=t_0\text{,}\)</span> we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_1.html ./knowl/eq8_3.html">
\begin{equation}
{\bf x}(t_0)={\bm \Psi}(t_0) \, {\bf c}.\tag{6.5.3}
\end{equation}
</div>
<p class="continuation">We note that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_1.html ./knowl/eq8_3.html">
\begin{equation*}
\det {\bm \Psi}(t)=\det [{\bf x}^{(1)}, {\bf x}^{(2)}, \cdots, {\bf x}^{(n)}]=W[{\bf x}^{(1)}, {\bf x}^{(2)}, \cdots, {\bf x}^{(n)}] \neq 0, \quad \alpha \leq t \leq \beta.
\end{equation*}
</div>
<p class="continuation">Thus, <span class="process-math">\({\bm \Psi}(t)\)</span> has an inverse <span class="process-math">\({\bm \Psi}^{-1}(t)\text{.}\)</span> Therefore, from (<a href="" class="xref" data-knowl="./knowl/eq8_3.html" title="Equation 6.5.3">(6.5.3)</a>),</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_1.html ./knowl/eq8_3.html">
\begin{equation*}
{\bf c}={\bm \Psi}^{-1}(t_0)\, {\bf x}(t_0).
\end{equation*}
</div>
<p class="continuation">The solution for the initial value problem is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_1.html ./knowl/eq8_3.html">
\begin{equation*}
{\bf x}={\bm \Psi}(t) \, {\bm \Psi}^{-1}(t_0)\, {\bf x}(t_0).
\end{equation*}
</div>
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